Outs calculator: the exact chance, and where the rule of four and two is wrong
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The starting numbers are the ones from the most familiar case, forty seven unseen and two still to come, but both are fields. The same arithmetic answers the same question in any card game, with any deck size.
Results
| Exact chance of hitting, in percent | — |
| What the rule of thumb says, in percent | — |
| How far off the rule is, in points | — |
| Chance with only the next card, in percent | — |
| What the price needs you to have, in percent | — |
| Margin over the price, in points | — |
The price line is the plainest arithmetic on the page. Putting in an amount to win a pot needs the amount divided by the pot plus the amount, so twenty five into a hundred needs twenty out of a hundred to break even over the long run, and nothing else about the hand changes that number.
Where the rule of thumb goes wrong
| Cards that help | Exact, in percent | Rule of thumb, in percent | Difference, in points |
|---|
The rule does not miss in one direction, it misses in two. With two cards still to come it comes in under the truth up to six helping cards and over it from seven onward, so the crossing is somewhere most hands actually live. Below the crossing the rule is being slightly pessimistic, and above it the error grows fast.
Read the last column downwards to see how fast. At nine it is a point, at fifteen it is nearly six, and at twenty it is twelve and a half. A player with a big draw who trusts the shortcut is being told about eighty out of a hundred when the truth is sixty seven and a half, which is the difference between a comfortable call and a bad one.
There is a bigger error than any of that, and it is not arithmetic. Multiplying by four assumes both remaining cards arrive for free, and that only happens when the money is already all in. If a bet is still coming, the honest number to compare against the current price is the one for a single card, which is the fourth row above and is roughly half as large.
None of this tells you to call or to fold. It puts an exact chance next to an exact price, and the parts it cannot see are the ones that usually decide: what else the pot might pay later, what the other player does next, and whether the cards that help you would really be enough.
How wrong is the rule of four and two?
It is wrong in two directions, which is the part the rule never mentions. With two cards still to come it lands under the true chance up to six helping cards and over it from seven onward.
Below the crossing the error is small and pessimistic. Above it the error grows fast, and by twenty helping cards the shortcut is overstating your chance by more than twelve points.
| Cards that help | Exact, in percent | Rule of thumb, in percent | Difference, in points |
|---|---|---|---|
| 2 | 8,42 | 8 | -0,42 |
| 6 | 24,14 | 24 | -0,14 |
| 7 | 27,84 | 28 | +0,16 |
| 9 | 34,97 | 36 | +1,03 |
| 12 | 44,96 | 48 | +3,04 |
| 15 | 54,12 | 60 | +5,88 |
| 20 | 67,53 | 80 | +12,47 |
When does multiplying by four not apply at all?
Whenever another bet is still coming. Multiplying by four counts both remaining cards, and you only get to see both of them for free if the money is already all in.
If you are paying now for one card and will be asked to pay again afterwards, the number to compare against the current price is the single-card one, which is roughly half as large. That is a bigger mistake than any rounding error in the shortcut itself.
How do I turn the money in the pot into a percentage?
Divide what you would put in by the pot plus what you would put in. That fraction is the chance you need just to break even in the long run, and nothing else about the hand moves it.
So a quarter of the pot asks for twenty out of a hundred, half the pot asks for thirty three, and a pot-sized bet asks for fifty. The bigger the price, the more of the deck has to be on your side.
| Pot | Amount to put in | Chance needed, in percent |
|---|---|---|
| 100 | 25 | 20,00 |
| 100 | 50 | 33,33 |
| 100 | 100 | 50,00 |
| 100 | 200 | 66,67 |
Does this work outside of poker?
Yes, and that is why the deck size is a field rather than a fixed number. The arithmetic only asks how many cards you cannot see, how many are still to come, and how many of them would help.
Any card game answers those three questions, so the same page serves a sixty-card deck as readily as a fifty-two card one. The starting values are just the most familiar case.
Does a good number here mean I should pay?
No. The page compares an exact chance with an exact price and stops there, because the things it cannot see are usually the ones that decide.
It does not know what the pot might pay you later, what the other player will do next, or whether the cards you are counting would really win the hand if they arrived. Counting a card that gives you a hand still worse than theirs is the most common way this arithmetic ends up correct and useless.
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